Question 674684: find the area of a triangle whose vertices are A(1, 1, 0) B(2, 1, 1) and C(1, 1, 2)
Answer by Edwin McCravy(20054)   (Show Source):
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A(1, 1, 0) B(2, 1, 1) and C(1, 1, 2)
The magnitude of the cross product of two vectors
is the area of the parallelogram which has adjacent
sides equal to the magnitudes of the two vectors.
We form the vectors AB and AC
AB = < 2-1, 1-1, 1-0 > = < 1, 0, 1 >
AC = < 1-1, 1-1, 2-0 > = < 0, 0, 2 >
We find their cross product AB×AC
 = [(0)(2)-(1)(0)]i - [(1)(2)-(1)(0)]j + [(1)(0)-(0)(0)]k
= 0i - (2-0)j + 0k = < 0, -2, 0 >
The magnitude of < 0, -2, 0 > is
√0² + (-2)² + 0² = √0 + 4 + 0 = √4 = 2
The triangle's area is  the area of that parallelogram.
Answer = (2) = 1
Edwin
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